grandes-ecoles

Q1 Matrices Linear Transformation and Endomorphism Properties View

Let $F$ be a vector subspace of $E$ stable under $u$. Show that the orthogonal complement $F^{\perp}$ of $F$ is stable under $u$.

Q2 Matrices Linear Transformation and Endomorphism Properties View

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for all real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Show that $\varphi$ is of class $\mathcal{C}^1$.

Q3 Matrices Linear Transformation and Endomorphism Properties View

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for all real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Calculate $\|\gamma(t)\|$ then justify that $\varphi'(0) = 0$.

Q4 Matrices Linear Transformation and Endomorphism Properties View

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for all real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.

Q5 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ Show that $x_0$ is an eigenvector of $u$.

Q6 Curve Sketching Sketching a Curve from Analytical Properties View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Let $s \in ]0,1[$. Sketch the graph of $k_s$ on $[0,1]$.

Q7 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ We also note, for all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$. Show that $K$ is continuous on $[0,1] \times [0,1]$.

Q8 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Show that $T$ is a continuous endomorphism of $E$.

Q9 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $F$ be the vector subspace of $E$ formed by polynomial functions. For $k \in \mathbb{N}$, we denote by $p_k$ the function defined by $p_k(x) = x^k$. For all $k \in \mathbb{N}$, calculate $T(p_k)$. Deduce that $F$ is stable under $T$.

Q10 Differential equations Integral Equations Reducible to DEs View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $F$ be the vector subspace of $E$ formed by polynomial functions. Deduce $(T(p))''$ for all $p \in F$.

Q11 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $f \in E$. Calculate $T(f)(0)$ and $T(f)(1)$.

Q12 Differential equations Integral Equations Reducible to DEs View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ For all $f \in E$, show that $T(f)$ is of class $\mathcal{C}^2$ then that $T(f)'' = -f$.

Q13 Differential equations Integral Equations Reducible to DEs View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Show that $T$ is injective.

Q14 Differential equations Integral Equations Reducible to DEs View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Determine the image of $T$.

Q15 Second order differential equations Second-order ODE with initial or boundary value conditions View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $\lambda \in \mathbb{R}$ be a nonzero eigenvalue of $T$ and $f$ be an associated eigenvector. Show that $f$ is a solution of the differential equation $\lambda f'' = -f$.

Q16 Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.

Q17 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. Justify that, for all $(f,g) \in E^2$, we have $$\langle T(f), g \rangle = \langle f, T(g) \rangle$$ One may use question 12.

Q18 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

Q19 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. Show that the family of vectors $(g_k)_{k \in \mathbb{N}^*}$ is orthonormal.

Q20 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2 \pi^2} \langle f, g_k \rangle g_k(x)$$ Show that $\Phi$ is continuous.

Q21 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2 \pi^2} \langle f, g_k \rangle g_k(x)$$ For all $N \in \mathbb{N}$, we set $f_N = \sum_{k=1}^N \langle f, g_k \rangle g_k$. Show that $$\lim_{N \rightarrow +\infty} \left\| T(f_N) - \Phi \right\| = 0$$

Q22 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2 \pi^2} \langle f, g_k \rangle g_k(x)$$ For all $N \in \mathbb{N}$, we set $f_N = \sum_{k=1}^N \langle f, g_k \rangle g_k$. Deduce $T(f) = \Phi$.

Q23 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. Show that we define an inner product on $E_1$ by setting $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$$

Q24 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. Show that, for every function $f : [0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2 \, \mathrm{d}t}$$

Q25 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t) \, \mathrm{d}t$$ Let $f \in E_1$ of class $\mathcal{C}^2$. Show that $U(f) = -T(f'')$. Deduce that $U(f) = f$.

Q26 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t) \, \mathrm{d}t$$ Show that $U$ is the identity map on $E_1$.

Q27 Integration by Parts Inner Product or Orthogonality Proof via Integration by Parts View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined in the previous part.

Q28 Integration by Parts Inner Product or Orthogonality Proof via Integration by Parts View

We consider the space $E$ of continuous functions from $[0,1]$ to $\mathbb{R}$, equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ Show that $(E, \langle \cdot, \cdot \rangle)$ is not a reproducing kernel Hilbert space.

Q29 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ be a sequence of reals such that the series $\sum (a_n)^2$ is convergent. Show that the radius of convergence of the power series $\sum a_n t^n$ is greater than or equal to 1.

Q30 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Show that $E_2$ equipped with $\langle \cdot, \cdot \rangle$ is a real pre-Hilbert space.

Q31 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Let $x \in ]-1,1[$. Determine $g_x \in E_2$ such that, for all $f \in E_2$, $$f(x) = \langle g_x, f \rangle$$

Q32 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Deduce that $E_2$ is a reproducing kernel Hilbert space and specify its kernel.

Q33 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

We are given a real $a > 0$. We consider the space $E_3$ of functions $f : [0,a] \rightarrow \mathbb{R}$, continuous and of class $\mathcal{C}^1$ piecewise on $[0,a]$, and satisfying $f(0) = 0$. We equip $E_3$ with the inner product defined, for $f, g \in E_3$, by $$(f \mid g) = \int_0^a f'(t) g'(t) \, \mathrm{d}t$$ Show that the function $(x,y) \mapsto \min(x,y)$ is a reproducing kernel on $(E_3, (\cdot \mid \cdot))$.

Q34 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

We are given a real $a > 0$. Let $E_4$ be the space of functions continuous on $[0,a]$, taking values in $\mathbb{R}$, of class $\mathcal{C}^1$ piecewise and furthermore satisfying $f(a) = 0$. Let $\varphi : [0,a] \rightarrow \mathbb{R}$ be of class $\mathcal{C}^1$ satisfying $\varphi(a) = 0$ and, for all $x \in [0,a]$, $\varphi'(x) < 0$. Determine an inner product on $E_4$ such that the function $(x,y) \mapsto \min(\varphi(x), \varphi(y))$ is a reproducing kernel on the pre-Hilbert space $E_4$.

Q35 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Let $x \in I$ and $V_x$ defined on $E$ by $V_x(f) = f(x)$. We set $$N(V_x) = \sup_{\|f\|=1} |f(x)|$$ Prove that $$N(V_x) = \sqrt{\langle k_x, k_x \rangle}.$$

Q36 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Suppose that $K$ is continuous on $I \times I$. Prove that all functions in $E$ are continuous.

Q37 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ We consider a function $A : [0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T : E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t) \, \mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. Justify that $T$ induces an isomorphism from $(\ker T)^\perp$ onto $\operatorname{Im} T$.

Q38 Proof Proof That a Map Has a Specific Property View

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ We consider a function $A : [0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T : E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t) \, \mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. We denote by $S$ the inverse bijection of the isomorphism induced by $T$ from $(\ker T)^\perp$ onto $\operatorname{Im} T$. We define the inner product $\varphi$ on $\operatorname{Im} T$ by setting, for all $(f,g) \in (\operatorname{Im} T)^2$, $$\varphi(f,g) = \langle S(f), S(g) \rangle$$ We consider the application $K$ defined on $[0,1]^2$ by $$K(x,y) = \int_0^1 A(x,t) A(y,t) \, \mathrm{d}t$$ Show that $(\operatorname{Im} T, \varphi)$ is a reproducing kernel Hilbert space, with kernel $K$.

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